The trace \(_B\)  on \({M}_{B}\)  can be described making use of the Dixmier trace along the line anticipated in [1]}.
There are several standard references
for the theory of the Dixmier trace, see, e.g., [2]}, [3]}, [4]}, [5]}, [6]}.
Here, we will recall only the basic definition of the Dixmier trace.
Let us start with the singular values \(\mu _n(T)\)  of a compact operator \(T\)  which are, by definition, the eigenvalues of \(|T|:=\sqrt{T^*T}\) . By convention the singular values will be listed in decreasing order, repeated according to their multiplicity, i.e.,
\(\mu _0(T) \geqslant \mu _1(T) \geqslant \cdots \geqslant \mu _n(T) \geqslant \mu _{n+1}(T) \geqslant \cdots \geqslant 0 .\) 
